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A New Class of Oscillatory Radial Basis Functions

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A New Class of Oscillatory Radial Basis Functions A NEW CLASS OF OSCILLATORY RADIAL BASIS FUNCTIONS BENGT FORNBERG ∗, ELISABETH LARSSON † , AND GRADY WRIGHT ‡ Abstract. Radial basis functions (RBFs) form a primary tool for multivariate interpolation, and they are also receiving increased attention for solving PDEs on irregular domains. Traditionally, only non-oscillatory radial functions have been considered. We find here that a certain class of oscillatory radial functions (including Gaussians as a special case) leads to non-singular interpolants with in- triguing features especially as they are scaled to become increasingly flat. This flat limit is important in that it generalizes traditional spectral methods to completely general node layouts. Interpolants based on the new radial functions appear immune to many or possibly all cases of divergence that in this limit can arise with other standard types of radial functions (such as multiquadrics and inverse multiquadratics). Key words. Radial basis functions, RBF, multivariate interpolation, Bessel functions. 1. Introduction. A radial basis function (RBF) interpolant of multivariate data (xk,yk), k =1, 2,...,n takes the form n s(x)= λ φ( x x ) . (1.1) k k − kk Xk=1 Here denotes the standard Euclidean vector norm, φ(r) is some radial function, and k·k an underline denotes that quantity to be a vector. The coefficients λk are determined in such a way that s(xk)= yk, k =1, 2,...,n, i.e. as the solution to the linear system λ1 y1 . . A . = . (1.2) λ y n n where the entries of the matrix A are A = φ x x , i = 1,...,n, ,j = i,j i − j 1,...,n. Numerous choices for φ(r) have been used in the past. Table 1 shows a few cases for which existence and uniqueness of the interpolants s(x) have been discussed in the literature; see for ex. [2], [3], [14], and [15]. For many of the radial functions in Table 1, existence and uniqueness are ensured for arbitrary point distributions. However, there are some that require the form of (1.1) to be augmented by some low-order polynomial terms. In the infinitely smooth cases, we have included a shape parameter ε in such a way that ε 0 corresponds to the basis functions becoming flat (as discussed extensively in for→ example [4], [7], [8], [11], [12]). The primary interest in this limit lies in the fact that it reproduces all the classical pseudospectral (PS) methods [6], such as Fourier, Chebyshev, and Legendre, whenever the data point locations are ∗University of Colorado, Department of Applied Mathematics, 526 UCB, Boulder, CO 80309, USA ([email protected]). The work was supported by NSF grants DMS-9810751 (VIGRE) and DMS-0309803. †Uppsala University, Department of Information Technology, Scientific Computing, Box 337, SE- 751 05 Uppsala, Sweden ([email protected]). The work was supported by a grant from The Swedish Research Council. ‡University of Utah, Department of Mathematics, 155 South 1400 East, JWB 233, Salt Lake City, UT 84112-0090, USA ([email protected]). The work was supported by NSF VIGRE grant DMS-0091675. 1 Type of basis function φ(r) Piecewise smooth RBFs Generalized Duchon spline (GDS) r2k log r, k N ∈ r2ν , ν > 0 and ν N 6∈ Wendland (1 r)k p(r), p a polynomial, k N − + ∈ 21 ν Mat´ern − rν K (r), ν > 0 Γ(ν) ν Infinitely smooth RBFs 2 (εr) Gaussian (GA) e− Generalized multiquadric (GMQ) (1 + (εr)2)ν/2, ν = 0 and ν 2N 6 6∈ Multiquadric (MQ) (1 + (εr)2)1/2 • 2 1/2 Inverse multiquadric (IMQ) (1 + (εr) )− • 2 1 Inverse quadratic (IQ) (1 + (εr) )− • Table 1.1 Some commonly used radial basis functions. Note: in all cases, ε> 0. distributed in a corresponding manner. The interpolant (1.1) can therefore be seen as a major generalization of the PS approach, allowing scattered points in arbitrary numbers of dimensions, a much wider functional choice, and a free shape parameter ε that can be optimized. The RBF literature has so far been strongly focused on radial functions φ(r) that are non-oscillatory. We are not aware of any compelling reason for why this needs to be the case. Although we will show that φ(r) oscillatory implies that the interpolation problem can become singular in a sufficiently high dimension, we will also show that this need not be of any concern when the dimension is fixed. The present study focuses on the radial functions J d (εr) 2 1 φd(r)= − , d =1, 2,..., (1.3) d 1 (εr) 2 − where Jα(r) denotes the J Bessel function of order α. For odd values of d, φd(r) can be alternatively expressed by means of regular trigonometric functions: 2 φ (r)= cos(εr) 1 rπ 2 sin(εr) φ (r)= 3 rπ εr 2 sin(εr) εr cos(εr) φ (r)= − 5 rπ (εr)3 . We will later find it useful to note that these φ (r) functions can also be ex- d − pressed in terms of the hypergeometric 0F1 function: 2 d 1 d φd(r)=2 − Γ 2 ψd(r) 2 1 d = 3 d = 5 0.8 d = 10 2 e− r 0.6 0.4 0.2 0 −0.2 −0.4 −8 −6 −4 −2 0 2 4 6 8 J √δr Fig. 1.1. Comparison between 2δδ! δ (2 ) for d = 3, 5, 10 (i.e. δ = 3 , 5 , 5) and the d →∞ (2√δr)δ 2 2 r2 limit e− . where ψ (r)= F ( d , 1 (εr)2). (1.4) d 0 1 2 − 4 In the d limit, the oscillations of φd(r) vanish, and Gaussian (GA) radial functions are→ recovered, ∞ as follows from the relation 2 δ Jδ(2√δr) r lim 2 δ! = e− . (1.5) δ √ δ →∞ (2 δr) Comparing the ratio above with (1.3), we have here written δ in place of d 1 and 2 − chosen ε = 2√δ. Figure 1.1 illustrates (1.5), comparing the curves for d = 3, 5, and 10 with the Gaussian limit. For these radial functions φd(r), we will prove non-singularity for arbitrarily scat- tered data in up to d dimensions (when d > 1). However, numerous other types of radial functions share this property. What makes the present class of Bessel-type basis functions outstanding relates to the flat basis function limit as ε 0. As a consequence of the limit (when it exists) taking the form of an interpolating→ poly- nomial, it connects pseudospectral (PS) methods [6] with RBF interpolants [8]. It was conjectured in [8] and shown in [16] that GA (in contrast to, say, MQ, IMQ, and IQ) will never diverge in this limit, no matter how the data points are located. The results in this study raise the question whether the present class of Bessel-type radial functions might represent the most general class possible of radial functions with this highly desirable feature. The radial functions φd(r) have previously been considered in [17] (where (1.5) and the positive semi-definiteness of the φd(r)-functions were noted), and in an example in [9] (in the different context of frequency optimization). They were also noted very 3 briefly in [8] as appearing immune to a certain type of ε 0 divergence – the main topic of this present study. → 2. Some observations regarding oscillatory radial functions. Expansions in different types of basis functions are ubiquitous in computational mathematics. It is often desirable that such functions are orthogonal to each other with regard to some type of scalar product. A sequence of such basis functions then needs to be increasingly oscillatory, as is the case for example with Fourier and Chebyshev functions. It can be shown that no such fixed set of basis functions can feature guaranteed non-singularity in more than 1-D when the data points are scattered [13]. The RBF approach circumvents this problem by making the basis functions dependent on the data point locations. It uses different translates of one single radially symmetric function, centered at each data point in turn. Numerous generalizations of this approach are possible (such as using different basis functions at the different data point locations, or not requiring that the basis functions be radially symmetric). The first question we raise here is why it has become customary to consider only non-oscillatory radial functions (with a partial exception being GDS φ(r)= r2k log r which changes sign at r = 1). One reason might be the requirements in the primary theorem that guarantees non-singularity for quite a wide class of RBF interpolants [3], [15]: Theorem 2.1. If Φ(r) = φ(√r) is completely monotone but not constant on [0, ), then for any points x in Rd, the matrix A in (1.2) is positive definite. ∞ k The requirement for φ(√r) to be completely monotone is far more restrictive than φ(r) merely being non-oscillatory: Definition 2.2. A function Φ(r) is completely monotone on [0, ), if (i) Φ(r) C[0, ) ∞ ∈ ∞ (ii) Φ(r) C∞(0, ) ∈ k ∞ (iii) ( 1)k d Φ(r) 0 for r> 0 and k =0, 1, 2,... − drk ≥ An additional result that might discourage the use of oscillatory radial functions is the following: Theorem 2.3. If φ(r) C[0, ) with φ(0) > 0 and φ(ρ) < 0 for some ρ > 0, then there is an upper limit∈ on the∞ dimension d for which the interpolation problem is non-singular for all point distributions. Proof. Consider the point distributions shown in Figure 2.1. The first row in the A matrix will have the d + 1 entries − [φ(0), φ(ρ), φ(ρ), φ(ρ),..., φ(ρ)] .
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